- The dimensional reduction modelling approach for 3D beams: Differential equations and finite-element solutions based on Hellinger–Reissner principle
- F. Auricchio ; G. Balduzzi ; C. Lovadina
- Book Title / Journal: International Journal of Solids and Structures
- Year: 2013 , Volume: 50 , Series:
- Structural Analysis
- Keywords: Linear elastic beam ; Mixed variational modelling ; Beam analytical solution ; Beam analytical solution Static analysis
- Description
- This paper illustrates an application of the so-called dimensional reduction modelling approach to obtain a mixed, 3D, linear, elastic beam-model.
We start from the 3D linear elastic problem, formulated through the Hellinger–Reissner functional, then we introduce a cross-section piecewise-polynomial approximation, and finally we integrate within the cross section, obtaining a beam model that satisfies the cross-section equilibrium and could be applied to inhomogeneous bodies with also a non trivial geometries (such as L-shape cross section). Moreover the beam model can predict the local effects of both boundary displacement constraints and non homogeneous
or concentrated boundary load distributions, usually not accurately captured by most of the
popular beam models.
We modify the beam-model formulation in order to satisfy the axial compatibility (and without violating equilibrium within the cross section), then we introduce axis piecewise-polynomial approximation, and finally we integrate along the beam axis, obtaining a beam finite element. Also the beam finite elements have the capability to describe local effects of constraints and loads. Moreover, the proposed beam finite element describes the stress distribution inside the cross section with high accuracy.
In addition to the simplicity of the derivation procedure and the very satisfying numerical performances, both the beam model and the beam finite element can be refined arbitrarily, allowing to adapt the model accuracy to specific needs of practitioners.
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- Mixed 3D Beam Models: Differential Equation Derivation and Finite Element Solutions
- F. Auricchio ; G. Balduzzi ; C. Lovadina
- Book Title / Journal: European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012)
- Year: 2012 , Volume: , Series:
- Structural Analysis
- Keywords: Hellinger-Reissner principle ; 3D beam model ; dimensional reduction ; mixed beam finite element
- Description
- In this document we illustrate the dimensional-reduction approach applied to 3D solid elastic equations in order to obtain a beam model.
We start from the Hellinger-Reissner (HR) principle, in a formulation which guarantees the selection of a compatible solution in a family of equilibrated fields. Then, we introduce a semidiscretization within the cross-section, this allows to reduce the problem’s dimension from 3D to 1D and to formulate the beam model. After a manipulation of the 1D weak model (done in order to guarantee the selection of an axis-equilibrated solution in a family of axis-compatible fields), we introduce a discretization along the beam axis obtaining the related beam Finite Element (FE).
On one hand, the initial HR principle formulation leads to an accurate stress analysis into the cross-section, on the other hand, the 1D model manipulation leads to an accurate displacement
analysis along the beam-axis. Moreover, the manipulation allows to statically condensate
stresses out at element level, improving the numerical efficiency of the FE algorithm.
In order to illustrate the capability of the method, we consider a slim cross-section beam that shows non trivial behaviour in bending and for which the analytical solution is available in literature. Numerical results are accurate in description of both displacement and stress variables, the FE solution converges to the analytical solution, and the beam FE models complex phenomena like anticlastic bending and boundary effects.
- Abstract
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- A new modeling approach for planar beams: finite-element solutions based on mixed variational derivations
- F. Auricchio ; G. Balduzzi ; C. Lovadina
- Book Title / Journal: Journal of Mechanics of Materials and Structures
- Year: 2010 , Volume: 5 , Series:
- Structural Analysis
- Keywords: planar beams ; Finite elements
- Description
- This paper illustrates a new modeling approach for planar linear elastic beams. Referring to existing
models, we first introduce the variational principles that could be adopted for the beam model derivation, discussing their relative advantages and disadvantages. Then, starting from the Hellinger–Reissner functional we derive some homogeneous and multilayered beam models, discussing some properties of their analytical solutions. Finally, we develop a planar beam finite element, following an innovative approach that could be seen as the imposition of equilibrium in the cross-section and compatibility along the axis. The homogeneous model is capable of reproducing the behavior of the Timoshenko beam, with the advantage that the shear correction factor appears naturally from the variational derivation; the multilayered beam is capable of capturing the local effects produced by boundary constraints and load distributions; the finite element is capable of predicting the cross-section stress distribution with high accuracy, and more generally the behavior of planar structural elements.
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